For centuries, an orderly view of planetary system architectures dominated the discourse on planetary systems. However, there is growing evidence that many planetary systems underwent a period of upheaval, during which giant planets "migrated" from where they formed. This thesis addresses a question key to understanding how planetary systems evolve: is planetary migration typically a smooth, disk-driven process or a violent process involving strong multi-body gravitational interactions? First, we analyze evidence from the dynamical structure of debris disks dynamically sculpted during planets' migration. Based on the orbital properties our own solar system's Kuiper belt, we deduce that Neptune likely underwent both planet-planet scattering and smooth migration caused by interactions with leftover planetesimals. In another planetary system, Beta Pictoris, we find that the giant planet discovered there must be responsible for the observed warp of the system's debris belt, reconciling observations that suggested otherwise. Second, we develop two new approaches for characterizing planetary orbits: one for distinguishing the signal of a planet's orbit from aliases, spurious signals caused by gaps in the time sampling of the data, and another to measure the eccentricity of a planet's orbit from transit photometry, "the photoeccentric effect." We use the photoeccentric effect to determine whether any of the giant planets discovered by the Kepler Mission are currently undergoing planetary migration on highly elliptical orbits. We find a lack of such "super-eccentric" Jupiters, allowing us to place an upper limit on the fraction of hot Jupiters created by the stellar binary Kozai mechanism. Finally, we find new correlations between the orbital properties of planets and the metallicity of their host stars. Planets orbiting metal-rich stars show signatures of strong planet-planet gravitational interactions, while those orbiting metal-poor stars do not. Taken together, the results of thesis suggest that suggest that both disk migration and planet-planet interactions likely play a role in setting the architectures of planetary systems.